Suppose that two treatments T1 and T2 of unknown efficacy are available and that a large number N of patients are to be treated. We begin by treating some number n of the N patients with T1 and another n with T2; then we use what appears to be the better treatment for the remaining N-2 2n patients. We define the "regret" to be the expected value of the difference (total benefit if we had used the better treatment on all n patients); (total benefit actually obtained by the proposed allocation of treatments). It is our object to minimize this regret by proper choice of the controllable parameter n, which may be any integer less than or equal to N/2. The choice of n must be made in ignorance of the true difference between the two treatments and of the random variation in response from one patient to another. Hence it is natural to consider a sequential determination of n. Methods for doing this have been proposed by F.J. Anscombe and T. Colton, both in the American Statistical Association Journal for June, 1963, but an evaluation of the performance characteristics of such methods is lacking because of severe difficulties in both the analytic and simulation approaches. We have done sufficient work to indicate that sequential methods are suprisingly effective in designing comparative clinical trials with a large patient horizon, and we propose to study this problem intensively with a view to obtaining more or less optimal designs both when the responses are continuous and when they are discrete.